Nelson-Aalen Estimator

Definition

Nelson-Aalen Estimatorfor cumulative hazard:

$\hat{H}(t)=\sum \_t\_i\le t\frac{d\_i}{Y\_i}$

where:

• $d\_i$ = number of events at time $t\_i$
• $Y\_i$ = number at risk just before $t\_i$

About Nelson-Aalen Approach

The Nelson-Aalen estimator calculates cumulative hazard by summing instantaneous hazard rates at each event time.

Step-by-step Intuition:

1. At time $t\_1$: Out of $Y\_1$ people at risk, $d\_1$ fail. $h(t\_1)=\frac{d\_1}{Y\_1}$

2. At time $t\_2$: Out of $Y\_2$ people at risk, $d\_2$ fail. $h(t\_2)=\frac{d\_2}{Y\_2}$

Cumulative hazard up to $t\_2$: $H(t\_2)=h(t\_1)+h(t\_2)=\frac{d\_1}{Y\_1}+\frac{d\_2}{Y\_2}$

Related

• Kaplan-Meier Estimator
• Cumulative Hazard